Weak distributive laws

Ross Street

Distributive laws between monads (triples) were defined by Jon Beck in the
1960s. They were generalized to monads in 2-categories and noticed to be
monads in a 2-category of monads. Mixed distributive laws are comonads in
the 2-category of monads; if the comonad has a right adjoint
monad, the mate of a mixed distributive law is an ordinary distributive
law. Particular cases are the entwining operators between algebras and
coalgebras. Motivated by work on weak
entwining operators, we define and
study a weak notion of distributive law for monads. In particular, each
weak distributive law determines a wreath product monad (in the
terminology of Lack and Street); this gives an advantage over the mixed
case.